A construction of cylindric and polyadic algebras from atomic relation algebras

Given a simple atomic relation algebra A and a finite n ≥ 3, we construct effectively an atomic n-dimensional polyadic equality-type algebra P such that for any subsignature L of the signature of P that contains the boolean operations and cylindrifications, the L-reduct of P is completely representable if and only if A is completely representable. If A is finite then so is P. It follows that there is no algorithm to determine whether a finite n-dimensional cylindric algebra, diagonal-free cylindric algebra, polyadic algebra, or polyadic equality algebra is representable (for diagonal-free algebras this was known). We also obtain a new proof that the classes of completely representable n-dimensional algebras of these types are nonelementary, a result that remains true for infinite dimensions if the diagonals are present, and also for infinite-dimensional diagonal-free cylindric algebras.